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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU Today’s Objectives: Students will be able to: 1.Apply the three equations of motion for a rigid body in planar motion. 2.Analyze problems involving translational motion. In-Class Activities: FBD of Rigid Bodies EOM for Rigid Bodies Translational Motion PLANAR KINETIC EQUATIONS OF MOTION: TRANSLATION

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU APPLICATIONS The boat and trailer undergo rectilinear motion. In order to find the reactions at the trailer wheels and the acceleration of the boat its center of mass, we need to draw the FBD for the boat and trailer. How many equations of motion do we need to solve this problem? What are they? =

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU APPLICATIONS (continued) As the tractor raises the load, the crate will undergo curvilinear translation if the forks do not rotate. If the load is raised too quickly, will the crate slide to the left or right? How fast can we raise the load before the crate will slide?

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU EQUATIONS OF TRANSLATIONAL MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. First, a coordinate system with its origin at an arbitrary point P is established. The x-y axes should not rotate and can either be fixed or translate with constant velocity. As discussed in Chapter 16, when a body is subjected to general plane motion, it undergoes a combination of translation and rotation.

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU If a body undergoes translational motion, the equation of motion is F = m a G. This can also be written in scalar form as F x = m(a G ) x and F y = m(a G ) y = In words: the sum of all the external forces acting on the body is equal to the body’s mass times the acceleration of it’s mass center. EQUATIONS OF TRANSLATIONAL MOTION (continued)

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU EQUATIONS OF ROTATIONAL MOTION We need to determine the effects caused by the moments of the external force system. The moment about point P can be written as (r i F i ) + M i = r ma G + I G M p = ( M k ) p = where r = x i + y j and M p is the resultant moment about P due to all the external forces. The term (M k ) p is called the kinetic moment about point P.

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU If point P coincides with the mass center G, this equation reduces to the scalar equation of M G = I G . In words: the resultant (summation) moment about the mass center due to all the external forces is equal to the moment of inertia about G times the angular acceleration of the body. Thus, three independent scalar equations of motion may be used to describe the general planar motion of a rigid body. These equations are: EQUATIONS OF ROTATIONAL MOTION (continued) F x = m(a G ) x F y = m(a G ) y and M G = I G or M p = (M k ) p

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU When a rigid body undergoes only translation, all the particles of the body have the same acceleration so a G = a and = 0. The equations of motion become: Note that, if it makes the problem easier, the moment equation can be applied about other points instead of the mass center. In this case, M A = (m a G ) d. EQUATIONS OF MOTION: TRANSLATION ONLY (Section 17.3) F x = m(a G ) x F y = m(a G ) y M G = 0

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU When a rigid body is subjected to curvilinear translation, it is best to use an n-t coordinate system. Then apply the equations of motion, as written below, for n-t coordinates. EQUATIONS OF MOTION: TRANSLATION ONLY (continued) F n = m(a G ) n F t = m(a G ) t M G = 0 or M B = e[m(a G ) t ] – h[m(a G ) n ]

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU PROCEDURE FOR ANALYSIS Problems involving kinetics of a rigid body in only translation should be solved using the following procedure: 1.Establish an (x-y) or (n-t) inertial coordinate system and specify the sense and direction of acceleration of the mass center, a G. 2.Draw a FBD and kinetic diagram showing all external forces, couples and the inertia forces and couples. 3.Identify the unknowns. 5.Remember, friction forces always act on the body opposing the motion of the body. 4.Apply the three equations of motion: F x = m(a G ) x F y = m(a G ) y F n = m(a G ) n F t = m(a G ) t M G = 0 or M P = (M k ) P

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU READING QUIZ 1.When a rigid body undergoes translational motion due to external forces, the translational equations of motion (EOM) can be expressed for _____________. A)the center of rotation B) the center of mass C)any arbitrary point D) All of the above. 2.The rotational EOM about the mass center of the rigid body indicates that the sum of moments due to the external loads equals _____________. A)I G B) m a G C) I G + m a G D) None of the above.

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU Given:A 50 kg crate rests on a horizontal surface for which the kinetic friction coefficient k = 0.2. Find:The acceleration of the crate if P = 600 N. Plan:Follow the procedure for analysis. Note that the load P can cause the crate either to slide or to tip over. Let’s assume that the crate slides. We will check this assumption later. EXAMPLE

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU The coordinate system and FBD are as shown. The weight of (50)(9.81) N is applied at the center of mass and the normal force N c acts at O. Point O is some distance x from the crate’s center line. The unknowns are N c, x, and a G. Applying the equations of motion: Solution: F x = m(a G ) x : 600 – 0.2 N c = 50 a G F y = m(a G ) y : N c – 490.5 = 0 M G = 0: -600(0.3) + N c (x)-0.2 N c (0.5) = 0 N c = 490 N x = 0.467 m a G = 10.0 m/s 2

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU Since x = 0.467 m < 0.5 m, the crate slides as originally assumed. If x was greater than 0.5 m, the problem would have to be reworked with the assumption that tipping occurred. EXAMPLE (continued)

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU CONCEPT QUIZ A B 1.A 2 lb disk is attached to a uniform 6 lb rod AB with a frictionless collar at B. If the disk rolls without slipping, select the correct FBD. 6 lb 2 lb NaNa NbNb FsFs 8 lb NaNa NbNb 6 lb 2 lb NaNa NbNb A) B)C) FsFs

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU CONCEPT QUIZ A B 2.A 2 lb disk is attached to a uniform 6 lb rod AB with a frictionless collar at B. If the disk rolls with slipping, select the correct FBD. 6 lb 2 lb NaNa NbNb k N a 8 lb NaNa NbNb A) B)C) 6 lb 2 lb NaNa NbNb s N a FkFk

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU GROUP PROBLEM SOLVING Given: A uniform connecting rod BC has a mass of 3 kg. The crank is rotating at a constant angular velocity of AB = 5 rad/s. Find:The vertical forces on rod BC at points B and C when = 0 and 90 degrees. Plan:Follow the procedure for analysis.

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU GROUP PROBLEM SOLVING (continued) Solution: Rod BC will always remain horizontal while moving along a curvilinear path. The acceleration of its mass center G is the same as that of points B and C. When = 0º, the FBD is: Applying the equations of motion: mr 2 = (3)(0.2)(5 2 ) G (3)(9.81) N CxCx CyCy BxBx 350mm ByBy Note that mr 2 is the kinetic force due to the body’s acceleration M C = (M k ) C (0.7)B y – (0.35)(3)(9.81) = -0.35(15) B y = 7.215 N F y = m(a G ) y C y + 7.215 – (3)(9.81) = -15 C y = 7.215 N

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU When = 90º, the FBD is: Applying the equations of motion: mr 2 G (3)(9.81) N CxCx CyCy BxBx 350 mm ByBy (3)(0.2)(5 2 ) M C = (M k ) C (0.7)B y – (0.35)(3)(9.81) = 0 B y = 14.7 N F y = m(a G ) y C y + 14.7 – (3)(9.81) = 0 C y = 14.7 N GROUP PROBLEM SOLVING (continued)

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“Dynamics by Hibbeler,” Dr. S. Nasseri, MET Department, SPSU ATTENTION QUIZ 2.The number of independent scalar equations of motion that can be applied to box A is A) OneB) Two C) ThreeD) Four 1.As the linkage rotates, box A undergoes A) general plane motion. B) pure rotation. C) linear translation. D) curvilinear translation. A 1.5 m = 2 rad/s

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